Find the Volume y=1-x^2 , y=0

Problem

y=1−x2,y=0

Solution

1. Identify the region and the axis of rotation. Since the task asks for the volume bounded by y=1−x2 and y=0 without specifying an axis, we assume rotation about the xaxis.

2. Find the limits of integration by setting the equations equal to each other to find the xintercepts.

1−x2=0

x2=1

x=−1,x=1

3. Apply the disk method formula for volume, which is V=(∫_a^b)(π*[ƒ(x)]2*d(x))

V=(∫_−1^1)(π*(1−x2)2*d(x))

4. Expand the integrand (1−x2)2

(1−x2)2=1−2*x2+x4

5. Integrate the expression with respect to x

V=π*(∫_−1^1)((1−2*x2+x4)*d(x))

V=π*[x−(2*x3)/3+(x5)/5]1(−1)

6. Evaluate the definite integral at the boundaries.

V=π*((1−2/3+1/5)−(−1+2/3−1/5))

V=π*(8/15−(−8/15))

V=(16*π)/15

Final Answer

V=(16*π)/15

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